More on Probability

The first probability in the last example, i.e., the fair 6-sided die, is a special case. It is called the equilikely case . For this example, it is the assignment of probabilities under the assumption that the die is fair. In real life, this assumption is a statistical hypothesis, which we may want to test. For example, you are playing craps for high stakes; hence, you may want to test to see if the die is fair. The die has only to be shaved slightly (loaded die) to change the probabilities. Of course the fourth example above is a die loaded to high numbers.

How would you test to see if the die is fair?

Lets answer one of the questions posed above. Suppose we roll two fair dice. What's the probability that the sum of the upfaces is 7 or 11? Here's the sample space again:

 
        6 -     *         *         *         *         *         *
  Die 2   -
          -
        5 -     *         *         *         *         *         *
          +
          -
          -
        4 -     *         *         *         *         *         *
          -
          +
        3 -     *         *         *         *         *         *
          -
          -
        2 -     *         *         *         *         *         *
          +
          -
        1 -     *         *         *         *         *         *
            ----+---------+---------+---------+---------+---------+--Die 1 
                1         2         3         4         5         6

Since the dice are fair, it seems that each of the points is equilikely. Since there are 8 (6 as "7" and 2 as "11") elements in the event of interest, the probability of a "7" or "11" is 8/36.

Note if we assume the equilikely case for assigning probabilities, then the probability of any event is just the number of elements in that event divided by the number of elements in the sample space.

Exercise 3.3.1  

1.

Six cards with the numbers 1 through 6 on them are well shuffled and two cards are dealt (without replacment). Find the probability that the sum of the numbers on the two cards is 7. Note order is not important here. For example, the hand with cards 1,2 in it is the same as the hand with cards 2,1. In the sample space there are 15 elements. List them. Then find the probability that the sum of the numbers on the two cards is 7. Why is your answer different from the craps game answer?